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A097862 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and height k (n>=0, k>=0). 3
1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 15, 5, 1, 31, 18, 1, 1, 63, 56, 7, 1, 127, 161, 33, 1, 1, 255, 441, 129, 9, 1, 511, 1170, 453, 52, 1, 1, 1023, 3036, 1485, 242, 11, 1, 2047, 7753, 4644, 990, 75, 1, 1, 4095, 19565, 14040, 3718, 403, 13, 1, 8191, 48930, 41392, 13145, 1872, 102 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Row sums are the Motzkin numbers (A001006).

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

FORMULA

The g.f. for column k is z^(2k)/[P_k*P_{k+1}], where the polynomials P_k are defined by P_0=1, P_1=1-z, P_k=(1-z)P_{k-1}-z^2*P_{k-2}.

EXAMPLE

Triangle begins:

1;

1;

1,1;

1,3;

1,7,1;

1,15,5;

1,31,18,1;

1,63,56,7;

1,127,161,33,1;

Row n contains 1+floor(n/2) terms.

T(5,2)=5 counts HUUDD, UUDDH, UUDHD, UHUDD and UUHDD, where U=(1,1), H=(1,0) and D=(1,-1).

MAPLE

P[0]:=1: P[1]:=1-z: for n from 2 to 10 do P[n]:=sort(expand((1-z)*P[n-1]-z^2*P[n-2])) od: for k from 0 to 8 do h[k]:=series(z^(2*k)/P[k]/P[k+1], z=0, 20) od: a:=proc(n, k) if k=0 then 1 elif n=0 then 0 else coeff(h[k], z^n) fi end: seq(seq(a(n, k), k=0..floor(n/2)), n=0..15);

# second Maple program:

b:= proc(x, y, m) option remember; `if`(y>x, 0,

      `if`(x=0, z^m, `if`(y>0, b(x-1, y-1, m), 0)+

       b(x-1, y, m)+b(x-1, y+1, max(m, y+1))))

    end:

T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):

seq(T(n), n=0..16);  # Alois P. Heinz, Mar 13 2017

MATHEMATICA

b[x_, y_, m_] := b[x, y, m] = If[y > x, 0, If[x == 0, z^m, If[y > 0, b[x - 1, y - 1, m], 0] + b[x - 1, y, m] + b[x - 1, y + 1, Max[m, y + 1]]]];

T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][ b[n, 0, 0]];

Table[T[n], {n, 0, 16}] // Flatten (* Jean-Fran├žois Alcover, May 12 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A001006.

Sequence in context: A114580 A257597 A097229 * A097612 A136011 A227984

Adjacent sequences:  A097859 A097860 A097861 * A097863 A097864 A097865

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Sep 01 2004

STATUS

approved

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Last modified January 20 02:38 EST 2018. Contains 297938 sequences.